Fill in the blank(s) to correctly complete each sentence. If ƒ(x) = 4^x, then ƒ(2) = and ƒ(-2) = ________.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form ƒ(x) = a^x, where 'a' is a positive constant and 'x' is the variable. These functions exhibit rapid growth or decay depending on the base 'a'. In this case, ƒ(x) = 4^x represents an exponential function where the base is 4, indicating that as 'x' increases, the value of ƒ(x) increases exponentially.

Evaluating Functions

Evaluating a function involves substituting a specific value for the variable in the function's expression. For example, to find ƒ(2) in the function ƒ(x) = 4^x, you replace 'x' with 2, resulting in ƒ(2) = 4^2 = 16. Similarly, for ƒ(-2), you substitute -2 for 'x', leading to ƒ(-2) = 4^(-2) = 1/16, demonstrating how to compute function values for both positive and negative inputs.

Properties of Exponents

Properties of exponents are rules that govern how to manipulate exponential expressions. Key properties include the product of powers (a^m * a^n = a^(m+n)), the power of a power ( (a^m)^n = a^(m*n)), and the negative exponent rule (a^(-n) = 1/a^n). Understanding these properties is essential for simplifying expressions and solving equations involving exponents, such as those found in the given function.